We present two universal hinge patterns that enable a strip of material to
fold into any connected surface made up of unit squares on the 3D cube
grid—for example, the surface of any polycube. The folding is
efficient: for target surfaces topologically equivalent to a sphere, the strip
needs to have only twice the target surface area, and the folding stacks at
most two layers of material anywhere. These geometric results offer a new way
to build programmable matter that is substantially more efficient than what is
possible with a square N × N sheet of material,
which can fold into all polycubes only of surface area O(N) and
may stack Θ(N2) layers at one point. We also show how
our strip foldings can be executed by a rigid motion without collisions
(albeit assuming zero thickness), which is not possible in general with 2D
sheet folding.

To achieve these results, we develop new approximation algorithms for milling
the surface of a grid polyhedron, which simultaneously give a
2-approximation in tour length and an 8/3-approximation in the number of
turns. Both length and turns consume area when folding a strip, so we build
on past approximation algorithms for these two objectives from 2D milling.